Ben Green (mathematician)

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For other people named Ben Green, see Ben Green (disambiguation).

Ben Green
Green in 2010
Born
Ben Joseph Green

(1977-02-27) 27 February 1977 (age 47)
Bristol, England
Alma materTrinity College, Cambridge
(BA, MMath, PhD)
AwardsClay Research Award (2004)
Salem Prize (2005)
Whitehead Prize (2005)
SASTRA Ramanujan Prize (2007)
EMS Prize (2008)
Fellow of the Royal Society (2010)
Sylvester Medal (2014)
Senior Whitehead Prize (2019)
Scientific career
FieldsMathematics
InstitutionsUniversity of Bristol
University of Cambridge
University of Oxford
Princeton University
University of British Columbia
Massachusetts Institute of Technology
Thesis Topics in Arithmetic Combinatorics  (2003)
Doctoral advisorTimothy Gowers
Doctoral studentsVicky Neale
Adam Harper

Ben Joseph Green

Waynflete Professor of Pure Mathematics at the University of Oxford
.

Early life and education

Ben Green was born on 27 February 1977 in

Senior Wrangler title. He stayed on for Part III and earned his doctorate under the supervision of Timothy Gowers, with a thesis entitled Topics in arithmetic combinatorics (2003). During his PhD he spent a year as a visiting student at Princeton University. He was a research Fellow at Trinity College, Cambridge between 2001 and 2005, before becoming a Professor of Mathematics at the University of Bristol from January 2005 to September 2006 and then the first Herchel Smith Professor of Pure Mathematics at the University of Cambridge from September 2006 to August 2013. He became the Waynflete Professor of Pure Mathematics at the University of Oxford on 1 August 2013. He was also a Research Fellow of the Clay Mathematics Institute and held various positions at institutes such as Princeton University, University of British Columbia, and Massachusetts Institute of Technology
.

Mathematics

The majority of Green's research is in the fields of

additive combinatorics, but he also has results in harmonic analysis and in group theory. His best known theorem, proved jointly with his frequent collaborator Terence Tao, states that there exist arbitrarily long arithmetic progressions in the prime numbers: this is now known as the Green–Tao theorem.[2]

Amongst Green's early results in additive combinatorics are an improvement of a result of Jean Bourgain of the size of arithmetic progressions in sumsets,[3] as well as a proof of the Cameron–Erdős conjecture on sum-free sets of natural numbers.[4] He also proved an arithmetic regularity lemma[5] for functions defined on the first natural numbers, somewhat analogous to the Szemerédi regularity lemma for graphs.

From 2004–2010, in joint work with

Hardy–Littlewood circle method. Many aspects of this theory, including the quantitative aspects of the inverse theorem for the Gowers norms,[7]
are still the subject of ongoing research.

Green has also collaborated with Emmanuel Breuillard on topics in group theory. In particular, jointly with Terence Tao, they proved a structure theorem[8] for approximate groups, generalising the Freiman-Ruzsa theorem on sets of integers with small doubling. Green also has worked, jointly with Kevin Ford and Sean Eberhard, on the theory of the symmetric group, in particular on what proportion of its elements fix a set of size .[9]

Green and Tao also have a paper[10] on algebraic combinatorial geometry, resolving the Dirac-Motzkin conjecture (see Sylvester–Gallai theorem). In particular they prove that, given any collection of points in the plane that are not all

collinear
, if is large enough then there must exist at least lines in the plane containing exactly two of the points.

Kevin Ford, Ben Green, Sergei Konyagin, James Maynard and Terence Tao, initially in two separate research groups and then in combination, improved the lower bound for the size of the longest gap between two consecutive primes of size at most .[11] The form of the previously best-known bound, essentially due to Rankin, had not been improved for 76 years.

More recently Green has considered questions in arithmetic Ramsey theory. Together with Tom Sanders he proved that, if a sufficiently large finite field of prime order is coloured with a fixed number of colours, then the field has elements such that all have the same colour.[12]

Green has also been involved with the new developments of Croot-Lev-Pach-Ellenberg-Gijswijt on applying the

Sárközy's theorem.[13]

Awards and honours

Green has been a Fellow of the Royal Society since 2010,[14] and a Fellow of the American Mathematical Society since 2012.[15] Green was chosen by the German Mathematical Society to deliver a Gauss Lectureship in 2013. He has received several awards:

References

  1. ^ Ben Green's results at International Mathematical Olympiad
  2. S2CID 1883951
    .
  3. S2CID 120755105
    .
  4. S2CID 119615076
    .
  5. S2CID 17451915
    .
  6. JSTOR 20752252
    .
  7. JSTOR 23350588
    .
  8. S2CID 119603959
    .
  9. S2CID 15188628
    .
  10. S2CID 15813230
    .
  11. arXiv:1412.5029 [math.NT
    ].
  12. S2CID 119140038
    .
  13. S2CID 119150134
    .
  14. ^ "- Royal Society".
  15. ^ List of Fellows of the American Mathematical Society. Retrieved 19 January 2013.
  16. ^ "List of LMS prize winners – London Mathematical Society".

External links

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